Xu, X., Wisnom, M. R., Li, X., & Hallett, S. R. (2015). A numerical … · 2017. 2. 15. · factor of up to 16. A Finite Element (FE) method using the explicit code LS-Dyna was applied

Xu, X., Wisnom, M. R., Li, X., & Hallett, S. R. (2015). A numericalinvestigation into size effects in centre-notched quasi-isotropic carbon/epoxylaminates. Composites Science and Technology, 111, 32-39. DOI:10.1016/j.compscitech.2015.03.001

Peer reviewed version

License (if available):CC BY-NC-ND

Link to published version (if available):10.1016/j.compscitech.2015.03.001

Link to publication record in Explore Bristol ResearchPDF-document

This is the accepted author manuscript (AAM). The final published version (version of record) is available onlinevia Elsevier at http://dx.doi.org/10.1016/j.compscitech.2015.03.001. Please refer to any applicable terms of useof the publisher.

University of Bristol - Explore Bristol ResearchGeneral rights

This document is made available in accordance with publisher policies. Please cite only the publishedversion using the reference above. Full terms of use are available:http://www.bristol.ac.uk/pure/about/ebr-terms.html

http://dx.doi.org/10.1016/j.compscitech.2015.03.001http://research-information.bristol.ac.uk/en/publications/a-numerical-investigation-into-size-effects-in-centrenotched-quasiisotropic-carbonepoxy-laminates(e3f4e408-6c1f-458c-8a98-f16e88bd02ef).htmlhttp://research-information.bristol.ac.uk/en/publications/a-numerical-investigation-into-size-effects-in-centrenotched-quasiisotropic-carbonepoxy-laminates(e3f4e408-6c1f-458c-8a98-f16e88bd02ef).html

* Corresponding author. Tel.: +44 (0)117 33 15775.

E-mail address: [email protected] (X. Xu)

A Numerical Investigation into Size Effects in Centre-Notched

Quasi-isotropic Carbon/Epoxy Laminates

Xiaodong Xua*, Michael R. Wisnoma, Xiangqian Lia, Stephen R. Halletta

a Advanced Composites Centre for Innovation & Science (ACCIS), University of Bristol,

University Walk, Bristol BS8 1TR, UK

ABSTRACT

Numerical modelling of scaled centre-notched [45/90/-45/0]4s carbon/epoxy

laminates was carried out. The in-plane dimensions of the models were scaled up by a

factor of up to 16. A Finite Element (FE) method using the explicit code LS-Dyna was

applied to study the progressive damage development at the notch tips. Cohesive

interface elements were used to simulate splits within plies and delaminations between

plies. A failure criterion based on Weibull statistics was used to account for fibre failure.

There is a good correlation between the numerical and experimental results, and the

scaling trend can be explained in terms of the growth of the notch tip damage zone. The

modelling gives new insights into the damage development in the quasi-isotropic

laminates with sharp cracks, specifically, the growth of splits, delaminations and local

fibre breakage.

Keywords: A. Laminate; B. Strength; C. Finite element analysis (FEA); C. Notch; Size

effect

mailto:[email protected]

2

1. Introduction

Notched tensile strength of composites is a critical design driver. For example,

notched tensile tests are important to determine the damage tolerance of composite

fuselage structures. The sizes of laboratory notched coupons are usually at the scale of

centimetres. In contrast, large composite structures are normally sized in metres. There

is an obvious dimension gap, so it is important to understand the relationship between

the notched tensile strength of the small coupons and that of the large structures.

A few numerical methods were used to investigate the size effects in notched

composite laminates [1-4]. These approaches did not simulate the detailed damage

development at the notches. As a result, they need additional fracture parameters (e.g.

trans-laminar fracture toughness) in order to capture the scaling of strength. In contrast,

this paper adopts a virtual test technique which simulates the detailed notch-tip damage

development at different load levels and in different specimen sizes.

Discrete transverse crack and delamination were found to be crucial mechanisms

in the failure of composite laminates [5]. For example, splitting and delamination can

significantly affect the stress gradient at the notch tip. Different numerical approaches

have been developed to study fracture and damage in composites, such as continuum

modelling [6-8], embedded crack modelling, e.g. eXtended Finite Element Method (X-

FEM) [9] and discrete modelling, e.g. cohesive interface methods [5, 10, 11]. Among

the above modelling techniques, numerical methods have been developed to simulate

matrix cracking and delamination initiating from free edges [12-14], and those initiating

from notches [9, 15, 16]. Compared with continuum damage modelling, cohesive

interface methods can better represent the physical mechanisms at the discontinuities

that arise at the discrete failures. There may be scope to apply the X-FEM approach in

3

the future, but that would require further development to combine with the Weibull

statistics based criterion for fibre breakage which is crucial in the current study.

A numerical technique using the explicit FE code LS-Dyna and cohesive interface

elements was developed to simulate the sub-critical damage in composite laminates

with open holes [17] and sharp cracks [18]. Such detailed modelling technique can

successfully predict the tensile strength of open-hole specimens and blocked-ply over-

height compact tension specimens. In those cases, the final failure follows immediately

from the first fibre failure. However, the dispersed-ply laminates with sharp cracks were

not well simulated, in which the first fibre failure does not lead to the final failure

straight away. Instead, a damage zone which consists of stable fibre breakage, multiple

splits and delaminations is observed at the crack tips [19]. Simulating the first fibre

failure alone is not enough, and simulation of the development of the damage zone and

its influence on progressive fibre failure is necessary for accurate predictions.

An experimental investigation into the size effects in in-plane scaled centre-

notched [45/90/-45/0]4s laminates has recently been conducted [19]. The damage zone

was shown to play an important role in the scaling of centre-notched tensile strength.

The centre-notched strength decreases towards a Linear Elastic Fracture Mechanics

(LEFM) asymptote as the notch length increases, with the size of the damage zone

approaching an approximately constant value. In the present paper, the damage

development in in-plane scaled centre-notched [45/90/-45/0]4s laminates was studied

through an FE approach based on that of Li et al. [18]. The sizes of the simulated

damage zones in the scaled models were compared with those in the CT images from

interrupted tests [19], which has not hitherto been done. Because the scaled FE models

simulate closely the damage zone behaviour, the size effects can be well represented

4

and explained in terms of the growth of the damage zone, which can be observed in

much greater detail than is usually possible experimentally. Using this information to

provide understanding of the mechanisms giving rise to the size effects in sharp notched

specimens is the main novelty of this paper.

2. Experimental specimen configuration modelled

A schematic of the in-plane scaled centre-notched specimens and their dimensions

are illustrated in Fig. 1. Detailed ply-by-ply 3D FE models with 8-node constant stress

solid elements are constructed in LS-Dyna. All nodes at its one end are fixed, with

uniform displacements applied to the nodes at the other end. Half thickness of each

specimen is modelled, with symmetric boundary conditions applied to the nodes at the

mid-plane. The in-plane dimensions of the quasi-isotropic specimens were scaled up by

a factor of up to 8. In addition, a larger specimen with only the width and notch length

doubled from the one-size-smaller specimens (named as the “short variant”) was also

modelled as a further comparison. FE analysis demonstrated that in the short variant

specimens the closer boundaries in the length direction do not affect the stress

distribution near the notches.

The material used in the tests was Hexcel HexPly® IM7/8552 carbon-epoxy pre-

preg with a nominal ply thickness of 0.125 mm. All specimens were of the same

[45/90/-45/0]4s layup. The nominal thickness was 4 mm, which is very close to the

actual specimen thickness of 4 mm (C.V. 1.4%).

3. FE model setup

3.1. Typical FE mesh

Fig. 2 illustrates a typical FE mesh. A triangular shaped sharp notch tip was

modelled. In the experiments, a 0.25 mm-wide notch tip was cut with a piercing saw

5

blade, which was proved to be sharp enough not to affect the measured fracture

toughness in Ref. [19]. The CT images from the experimental study show that fibre

breakage is usually constrained within ±45° lines starting from the notch tips. So in the

FE analysis, a refined mesh was arranged near the notch tips within the ±45° lines to be

able to simulate the progressive damage development. A coarser mesh was used outside

this region.

The models were set up with a nominal ply thickness of 0.125 mm, so have a

thickness of 4 mm, similar to the measured value of 4 mm (C.V. 1.4%). The model of

the baseline specimens with one element through the thickness of each ply was

compared with a model with two elements through the thickness of each ply. The results

were within 1.2%, so only one element through each ply thickness was used in all of the

subsequent FE models.

3.2. Cohesive interface elements

In the FE analysis, cohesive interface elements were used to simulate the splits

within plies and the delaminations between plies. Specifically, to simulate the damage

zone at the notch tips, multiple potential split paths in the 0° plies were pre-defined. For

example, there are 9 pre-defined potential 0° split paths (marked in red) in the typical

FE mesh in Fig. 2 (a). In contrast, there is only a single pre-defined potential split path,

starting from each notch tip, in the plies with other orientations (±45° and 90°). This is

because the models showed that there is no fibre breakage in the other plies before final

failure, and no further potential split paths are needed to blunt the stress concentrations

after initial fibre fracture. Additional potential split paths could have been included in

these plies. However they would not affect the results and would increase computation

time. Fig. 2 (b) illustrates how the potential split paths are arranged. The properties of

6

the cohesive interface elements are shown in Table 1 [17].The mixed-mode traction

displacement relationship for cohesive interface elements is shown in Fig. 3 [20].

3.3. Fibre failure criterion

A criterion based on Weibull statistics has been used to predict fibre failure. The

theory supposes that the strength of a brittle-like material is controlled by defects which

follow a Weibull distribution, and the strength is related to the stressed volume [21].

When the volume adjusted stress reaches the unnotched unidirectional strength, fibre

failure will occur. Using the assumption of equal probability of survival between the

model and unit volume of material, we have Equation 1 [18]:

(1)

where, ơi is the elemental stress, Vi is the elemental volume, ơunit = 3131 MPa is the

tensile strength of a unit volume of material, m = 41 is the Weibull modulus from scaled

unnotched unidirectional tensile tests of the same material [22]. Other lamina properties

are shown in Table 1 [17].

Equation 1 is checked at each time step. When this fibre failure criterion is satisfied, the

element with the maximum fibre direction stress loses its load carrying capability and

its contribution is removed. After this, the load is automatically redistributed to the

other remaining elements. With increasing applied load, the stresses in the remaining

elements keep increasing until Equation 1 is satisfied again, then the next element with

the maximum fibre direction stress is degraded. This represents the continuous fibre

breakage process within the damage zone [18]. The same modelling technique was

successfully used to study progressive damage in over-height compact tension models

[18]. It was found that once the local fibre breakage in the 0° plies initiates, it does not

1)()(Elements Solid of No. Total

1 unitunit

i

i

mi

V

m VdV

7

propagate unstably. This is because although the first fibre breakage occurs due to the

high Stress Concentration Factor (SCF) at the notch tip, there is not enough energy for it

to propagate. Instead, it is arrested and secondary splits start to grow ahead of it. In the

current FE analysis, Equation 1 is insufficient on its own, and needs to work together

with multiple potential 0° split paths in order to simulate such damage zone behaviour.

4. FE analysis

4.1. Typical damage development

In the model of the baseline specimens (C = 3.2 mm) with 0.1 mm minimum

mesh size, 9 potential split paths were pre-defined within a 1 mm distance from the

notch tips in order to investigate the damage development at different load levels. The

degraded elements representing fibre breakage are marked in black. The fully failed

cohesive interface elements in which the critical strain energy release rate has been

exceeded are marked in red, corresponding to splits and delaminations. As shown in Fig.

4, the damage develops in the following sequence: At the beginning, initial splits grow

with applied stress in all the plies. Delamination then starts to occur at a higher stress

level and splits with different orientations can join up. As the applied stress increases,

fibre failure occurs in some 0° plies, at which point further delaminations also occur.

Because the simulation is under displacement control, there is no external work done

when fibres break. As the stored elastic energy is released, there is a load drop and the

newly formed crack arrests. When the applied stress increases again, secondary 0° splits

grow at the new crack front. The average distance between the last newly formed splits

and the notch tip is measured as the size of the damage zone. After a certain amount of

damage development, the fibre breakage propagates unstably across the model width,

which terminates the simulation. This corresponds to the final failure in the tests and the

8

final load drops on the load-displacement curves in Fig. 5 (d) and Fig. 6 (c). Because of

the unstable nature of the specimens, the damage status beyond 95% of failure load is

extremely hard to capture experimentally. However, the modelling is able to show the

stable damage development prior to catastrophic fracture which is key to understanding

size effects.

4.2. Mesh dependency

The above mesh for the baseline specimens (0.1 mm) is doubled to form the Scale

2 mesh (0.2 mm). Different densities of the potential 0° split paths in the Scale 2 model

are compared in Fig. 5. The FE results are illustrated in Fig. 5 (d), and were found not to

be sensitive to the density of potential split paths for a spacing of 1 mm or less. The

Scale 2 mesh was refined at the notch tips to form the fine mesh (0.1 mm) as shown in

Fig. 6. The results in Fig. 6 (c) show a reduction only of 3.2% for the refined mesh. So

the Scale 2 mesh (0.2 mm) is considered to be sufficiently refined. The results are not

sensitive to the mesh sizes, because the predicted splits blunt the stress concentration at

the notch tip. Although the model of the baseline specimen used a smaller 0° split

spacing (0.125 mm split spacing), in order to better capture the critical size of the

damage zone, the Scale 2 mesh (0.2 mm) with 1 mm split spacing was chosen as the

standard mesh and was used for the scaled up models. The Scale 4, Scale 8 and Scale 16

models have approximately the same minimum absolute mesh size (0.2 mm), and a split

spacing of 1 mm covering a distance of 4 mm from the crack tip. In the Scale 8 and

Scale 16 models, the mesh at the crack tip is refined over a larger area in order to

capture the slightly larger damage zone found subsequently in the analysis.

In other more general cases, the pre-defined multiple potential 0° split paths

should cover a distance that is larger than the size of the fully developed damage zone in

9

the specimen. If in the analysis damage grows beyond this distance, then it needs to be

extended. Initially, it is better to extend that distance across the whole width of the

specimen with 1 mm or less spacing, which could later be reduced once the damage

zone size is determined from the model.

4.3. Scaling of damage zones

In Fig. 7, the damage zones in the scaled FE models are compared with those in

the CT images from interrupted tests, each at the same load, which is 95% of the mean

experimental failure loads for each different specimen size. The FE models with pre-

defined multiple potential 0° split paths can simulate the stable fibre failure propagation

within the damage zone, the delamination shapes, split lengths and the trend for the

increase in damage zone as a function of notch size. The size effects, i.e. the scaling of

tensile strengths should therefore be able to be predicted. It can also be seen in Fig. 7

that the damage zones observed from both the interrupted tests and the FE models are

approaching an asymptote.

In Fig. 8, the sub-critical damage and fibre breakage in the scaled FE models are

compared at both approximately constant applied stress (340 MPa) and approximately

constant strain energy release rate G (30 kJ/m2, well below the value corresponding to

fracture energy due to fibre failure) for different specimen sizes. G is calculated

according to Equation 2 [23], which is valid for quasi-isotropic laminates ignoring ply

level effects such as free edge stresses and damage. Equation 2 cannot be directly

applied to anisotropic laminates. Although Laffan et al.[24] applied a similar equation

for orthotropic laminates, they did not recommend applying it to highly orthotropic

laminates.

10

E

CfG

2

)(2

g

2 (2)

where, G is the strain energy release rate, ơg is the applied gross section stress,

025.1)sec()( f is a geometric parameter to account for the effect of finite

width [25], C is the initial full notch length, W is specimen width, 1.02/ WC and

E = 61.6 GPa is the in-plane Young’s modulus.

The stress and G values are not exactly constant due to finite output time steps

used in the explicit analysis resulting in discrete values. The extent of damage is

approximately the same at constant G, but increases with size at constant applied stress,

which indicates that the development of the damage zone is driven by energy.

4.4. Propagation of initial 0° splits

With the available FE data, the development of damage in the scaled models was

also examined in terms of the growth of the initial 0° splits at the notch tips. The FE

analysis in this section is based on a simpler model, which only includes the initial

potential split paths, and no fibre failure criterion. The minimum mesh size is kept the

same (0.2 mm) for the scaled models.

In Fig. 9, the initial 0° split lengths increase linearly with increasing G, which

implies that the initial 0° splits are driven by energy. Splits are longer in the central

double 0° plies than those in the outboard single 0° plies.

The SCFs in the 0° plies in the simpler model of the baseline specimens were

studied in Fig. 10 (a). The SCF is calculated by using the maximum elemental stress

divided by the applied gross section stress. The SCFs decrease linearly with increasing

applied stress. Fig. 10 (a) also illustrates that the central double 0° ply has a lower SCF

at the same applied stress, which explains why the fibres in the central double 0° plies

11

break at a higher applied stress level. The reason for the lower SCF in the central double

0° ply is that the splits are longer than those in the single 0° plies at the same stress level

as shown in Fig. 9, due to more energy being available in the thicker ply block.

The 0° split length at the notch tip in the scaled models is normalised by the notch

length in Fig. 10 (b), and the SCFs also decrease with increasing normalised split length.

This explains why the SCFs decrease with increasing applied stress, because the initial

0° splits grow with applied stress (or G) as shown in Fig. 9, which can further blunt the

stress concentration at the notch tip. Fig. 10 (b) also illustrates that the central double 0°

ply has a slightly lower SCF even at the same split length, because it is affected by the

split lengths in the other plies.

4.5. Result comparison

There is a good correlation between the numerical and the experimental results as

shown in Fig. 11. The FE result for the baseline specimens (583 MPa) is spot on, the

Scale 2 specimens (542 MPa) 4.4% high, the Scale 4 specimens (489 MPa) 7.2% high,

the Scale 8 specimens (412 MPa) 18.1% high and the Scale 16 specimens (291 MPa)

11.5% high, with an average difference of 8.3% for the whole set.

In Fig. 11, the LEFM scaling line is determined from the constant GScale16 = 115.2

kJ/m2 calculated from Equation 2 by using the predicted tensile strength σg,Scale16 = 291

MPa of the largest Scale 16 model. The results for the smaller specimens are below this

line, but as the notch lengths increase, the tensile strengths from the larger models

clearly approach the LEFM scaling line. This is consistent with the experimental study

[19], and agrees with the scaling trend of the predicted damage zones which approach

an approximately constant size as the simulated specimens get larger. Unfortunately it is

12

not computationally practical to run even larger models, and this would be of limited

value since there are no experimental results to compare with.

5. Discussion

The Scaled 8 and Scale 16 models have approximately the same minimum mesh

size (0.2 mm) at the notch tips to that in the Scale 2 model. However, the mesh away

from the notch tips is coarser than that in the smaller models in order to make the

models computationally efficient. As a result, the unstable fracture beyond the damage

zone is sometimes artificially delayed when it propagates into the coarse mesh in the

larger models, which contributes to the larger discrepancy between the numerical and

experimental results.

The propagation of the initial 0° splits is driven by energy, as demonstrated by Fig.

9. Similarly, the development of the damage zone as a whole is driven by energy, as

seen in Fig. 8. The strength of the brittle fibres follows a Weibull statistics based limit.

The growth of initial splits can delay fibre failure by reducing the SCFs as shown in Fig.

10 (b) (stress blunting). Stable fibre breakage in the 0° plies can release the strain

energy at the notch tips and form a damage zone, followed by arrest of the initial crack

propagation and initiation of secondary splits. The failure criterion for the sub-critical

damage and that for the fibres interact with each other, resulting in the observed size

effects in the notched laminates. Specifically, for the smaller models, although their

strengths are higher, the energy levels at final failure are actually lower. For example,

according to Equation 2, the model of the baseline specimens predicts failure at σg,Baseline

= 583 MPa, so at GBaseline = 29.1 kJ/m2. In contrast, the Scale 16 model predicts failure

at σg,Scale16 = 291 MPa, so at GScale16 = 115.2 kJ/m2. For the smaller models, the damage

zones are under-developed due to lower G, and the energetic and statistical scaling laws

13

both contribute to the size effects, which results in an intermediate scaling trend. For the

larger models, the damage zones are fully developed due to higher G, so the energetic

size effect dominates, with the strength scaling approaching an LEFM scaling line and

the damage zone approaching an approximately constant size.

6. Conclusions

The present detailed modelling using cohesive interface elements provides a

powerful numerical tool for understanding the progressive damage development at the

notch tips of the centre-notched quasi-isotropic laminates. It can simulate the interaction

among different failure mechanisms within the damage zone such as splitting,

delamination and fibre breakage. The FE results are not dependent on the mesh or split

density provided these are fine enough and the pre-defined potential 0° split paths

extend over a distance larger than the fully developed damage zone.

The development of the damage zone is studied through the detailed modelling.

The present scaled FE models do not show catastrophic failure when the fibre breakage

initiates. Instead, complete failure only occurs after a period of progressive fibre

breakage in the 0° plies within the damage zone, which agrees with the experimental

observations.

There is a good correlation between the numerical and experimental results. This

is because the detailed FE modelling can represent the delamination shapes, split

lengths and the scaling of the damage zone as a function of notch size. With the pre-

defined multiple potential 0° split paths at the notched tips and the Weibull criterion for

fibre failure, the stable fibre failure propagation within the damage zone can also be

simulated. As the notch lengths increase, the tensile strengths are predicted to decrease

towards an LEFM scaling line, which is consistent with the experimental study.

14

References

[1] Cairns DS, Ilcewicz LB, Walker T, Minguet PJ. Fracture scaling parameters of

inhomogeneous microstructure in composite structures. Composites Science and

Technology. 1995;53(2):223-231.

[2] Gonzáles L, Knauss WG. Scaling global fracture behavior of structures-sized

laminated composites. International Journal of Fracture. 2002;118(4):363-394.

[3] Rudraraju SS, Salvi A, Garikipati K, Waas AM. In-plane fracture of laminated fiber

reinforced composites with varying fracture resistance: Experimental observations and

numerical crack propagation simulations. International Journal of Solids and Structures.

2010;47(7-8):901-911.

[4] Camanho PP, Erçin GH, Catalanotti G, Mahdi S, Linde P. A finite fracture

mechanics model for the prediction of the open-hole strength of composite laminates.

Composites Part A: Applied Science and Manufacturing. 2012;43(8):1219-1225.

[5] Wisnom MR. Modelling discrete failures in composites with interface elements.

Composites Part A: Applied Science and Manufacturing. 2010;41(7):795-805.

[6] Chang FK, Chang KY. A progressive damage model for laminated composites

containing stress concentrations. Journal of Composite Materials. 1987;21(9):834-855.

[7] Davila CG, Camanho PP, Rose CA. Failure criteria for FRP laminates. Journal of

Composite Materials. 2005;39(4):323-345.

[8] Forghani A, Zobeiry N, Vaziri R, Poursartip A, Ellyin F. A non-local approach to

simulation of damage in laminated composites. Damage in Composites. 2013;5:279-294.

[9] Mollenhauer D, Iarve EV, Kim R, Langley B. Examination of ply cracking in

composite laminates with open holes: A moiré interferometric and numerical study.

Composites Part A: Applied Science and Manufacturing. 2006;37(2):282-294.

15

[10] Pinho ST, Iannucci L, Robinson P. Formulation and implementation of decohesion

elements in an explicit finite element code. Composites Part A: Applied Science and

Manufacturing. 2006;37(5):778-789.

[11] Turon A, Camanho PP, Costa J, Renart J. Accurate simulation of delamination

growth under mixed-mode loading using cohesive elements: Definition of interlaminar

strengths and elastic stiffness. Composite Structures. 2010;92(8):1857-1864.

[12] Rybicki EF, Schmueser DW, Fox J. An energy release rate approach for stable

crack growth in free-edge delamination problem. Journal of Composite Materials.

1977;11:470-487.

[13] O'Brien TK, Hooper SJ. Local delamination in laminates with angle ply matrix

cracks, Part I: tension tests and stress analysis. Composite Materials: Fatigue and

Fracture, ASTM STP 1156. 1993;4:491-506.

[14] Soutis C, Kashtalyan M. Delamination growth and residual properties of cracked

orthotropic laminates under tensile loading. Journal of Thermoplastic Composite

Materials. 2002;15(1):13-22.

[15] Kortschot MT, Beaumont PWR. Damage mechanics of composite materials: II-A

damaged-based notched strength model. Composites Science and Technology.

1990;39(4):303-326.

[16] Wisnom MR, Hallett SR. The role of delamination in strength, failure mechanism

and hole size effect in open hole tensile tests on quasi-isotropic laminates. Composites

Part A: Applied Science and Manufacturing. 2009;40(4):335-342.

[17] Hallett SR, Green BG, Jiang WG, Wisnom MR. An experimental and numerical

investigation into the damage mechanisms in notched composites. Composites Part A:

Applied Science and Manufacturing. 2009;40(5):613-624.

16

[18] Li X, Hallett SR, Wisnom MR. Numerical investigation of progressive damage and

the effect of layup in overheight compact tension tests. Composites Part A: Applied

Science and Manufacturing. 2012;43(11):2137-2150.

[19] Xu X, Wisnom MR, Mahadik Y, Hallett SR. An experimental investigation into

size effects in quasi-isotropic carbon/epoxy laminates with sharp and blunt notches.

Composites Science and Technology. 2014;100(0):220-227.

[20] Jiang W-G, Hallett SR, Green BG, Wisnom MR. A concise interface constitutive

law for analysis of delamination and splitting in composite materials and its application

to scaled notched tensile specimens. International Journal for Numerical Methods in

Engineering. 2007;69(9):1982-1995.

[21] Wisnom MR. Size effects in the testing of fibre-composite materials. Composites

Science and Technology. 1999;59(13):1937-1957.

[22] Wisnom MR, Khan B, Hallett SR. Size effects in unnotched tensile strength of

unidirectional and quasi-isotropic carbon/epoxy composites. Composite Structures.

2008;84(1):21-28.

[23] Edwalds HL, Wanhill RJH. Fracture mechanics: Edward Arnold, Delftse Uitgevers

Maatschappij 1984.

[24] Laffan MJ, Pinho ST, Robinson P, Iannucci L. Measurement of the in situ ply

fracture toughness associated with mode I fibre tensile failure in FRP. Part I: Data

reduction. Composites Science and Technology. 2010;70(4):606-613.

[25] Newman JC. Fracture analysis of various cracked configurations in sheet and plate

materials. Properties Related to Fracture Toughness ASTM STP 605. 1976:104-123.

17

Fig. 1. Schematic of the in-plane scaled centre-notched specimens and dimensions (mm).

Fig. 2. Typical FE mesh.

18

Fig. 3. Mixed-mode traction displacement relationship for cohesive interface elements.

Fig. 4. Damage development in the model of baseline specimens (All layers

superimposed).

19

Fig. 5. Split density effects in the 0° plies in the Scale 2 model.

Fig. 6. Mesh size effects in the Scale 2 model.

20

Fig. 7. Damage zone comparison at 95% of the mean experimental failure load.

Fig. 8. Damage zone comparison at approximately constant applied stress and

approximately constant G.

21

Fig. 9. Development of the initial splits in the 0° plies in scaled simpler models.

Fig. 10. Development of SCFs in the 0° plies in the simpler model of baseline

specimens.

22

Fig. 11. Comparison of experimental and FE results.

Table 1. Properties of cohesive interface elements and lamina elements.

Properties of cohesive interface elements

GIC (N/mm) GIIC (N/mm) σImax (MPa) σIImax (MPa)

0.2 1.0 60 90

Properties of lamina elements

E11 (GPa) E22=E33 (GPa) G12=G13 (GPa) G23 (GPa) Weibull modulus m

161 11.4 5.17 3.98 41

σ11max (MPa) α22= α33 (⁰C-1) α11 (⁰C-1) υ12=υ13 υ23

3131* 310-5 0.0 0.320 0.436

* 3131 MPa is for unit volume material.